Solid State Physics

MAGNETISM ILecture 27

A.H. HarkerPhysics and Astronomy

UCL

10 Magnetic Materials

Magnet technology has made enormous advances in recent years –without the reductions in size that have come with these advancesmany modern devices would be impracticable.

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The important quantity for many purposes is the energy density ofthe magnet.

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10.1 Magnetic properties - reminder

There are two fields to consider, the magnetic fieldH which is gener-ated by currents according to Amp̀ere’s law, and the magnetic induc-tion, or magnetic flux density,B, which gives the torque experiencedby a dipole momentm asG = m×B. H is measured inA m−1 (Oer-steds in old units),B in Wb m−2 or T (Gauss in old units). In freespace,B = µ0H. In a material

B = µ0(H +M)

= µ0µrH= µ0(1 + χ)H,

whereµr is the relative permeability, χ is the magnetic susceptibility,which is a dimensionless quantity.

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Note, though, thatχ is sometimes tabulated as themolar susceptibility

χm = Vmχ,

whereVm is the volume occupied by one mole, or as themass suscep-tibility

χg =χ

ρ,

where ρ is the density.M, the magnetisation, is the dipole momentper unit volume.

M = χH.

In general, µr (and henceχ) will depend on position and will be ten-sors (so thatB is not necessarily parallel toH). Even worse, somematerials are non-linear, so thatµr and χ are field-dependent.

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The effects arehighly exaggeratedin these diagrams.6

10.2 Measuring magnetic properties

10.2.1 Force method

Uses energy of induced dipole

E = −1

2mB = −1

2µ0χVH2,

so in an inhomogeneous field

F = −dE

dx=

1

2µ0V χ

dH2

dx= µ0V χHdH

dx.

Practically:

• set up large uniformH;

• superpose linear gradient with additional coils

• vary second field sinusoidally and use lock-in amplifier to measurevarying force

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10.2.2 Vibrating Sample magnetometer

• oscillate sample up and down

• measure emf induced in coils A and B

• compare with emf in C and D from known magnetic moment

• hence measured sample magnetic moment8

10.3 Experimental data

In the first 60 elements in the periodic table, the majority have nega-tive susceptibility – they arediamagnetic.

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10.4 Diamagnetism

Classically, we have Lenz’s law, which states that the action of a mag-netic field on the orbital motion of an electron causes a back-emfwhich opposes the magnetic field which causes it. Frankly, this isan unsatisfactory explanation, but we cannot do better until we havestudied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector potentials. Imagine an electron in an atom asa chargee moving clockwise in the x-y plane in a circle of radiusa,areaA, with angular velocity ω. This is equivalent to a current

I = charge/time = eω/(2π),

so there is a magnetic moment

µ = IA = eωa2/2.

The electron is kept in this orbit by a central force

F = meω2a.

Now if a flux density B is applied in the z direction there will be aLorentz force giving an additional force along a radius

∆F = evB = eωaB10

If we assume the charge keeps moving in a circle of the same radius (avague nod towards quantum mechanics?) it will have a new angularvelocity ω′,

meω′2a = F −∆F

someω

′2a = meω2a− eωaB,

orω′2 − ω2 = −eωB

me.

If the change in frequency is small we have

ω′2 − ω2 ≈ 2ω∆ω,

where∆ω = ω′ − ω. Thus

∆ω = − eB2me

,

the Larmor frequency. Substituting back into

µ = IA = eωa2/2.

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we find a change in magnetic moment

µ = −e2a2

4meB.

Recall that a was the radius of a ring of current perpendicular to thefield: if we average over a spherical atom

a2 = 〈x2〉 + 〈y2〉 =2

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[〈x2〉 + 〈y2〉 + 〈z2〉

]=

2

3〈r2〉,

so

µ =e2〈r2〉6me

B,

and finally, if we have n atoms per volume, each withp electrons inthe outer shells, the magnetisation will be

M = npµ,

and

χ =MH

= µ0MB

= −µ0npe2〈r2〉6me

.

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Values of atomic radius are easily calculated: we can confirm thep〈r2〉 dependence.

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Diamagnetic susceptibility:

• Negative

• Typically −10−6 to −10−5

• Independent of temperature

• Always present, even when there are no permanent dipole mo-ments on the atoms.

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10.5 Paramagnetism

Paramagnetism occurs when the material contains permanent mag-netic moments. If the magnetic moments do not interact with eachother, they will be randomly arranged in the absence of a magneticfield. When a field is applied, there is a balance between the internalenergy trying to arrange the moments parallel to the field and en-tropy trying to randomise them. The magnetic moments arise fromelectrons, but if we they are localised at atomic sites we can regardthem as distinguishable, and use Boltzmann statistics.

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10.5.1 Paramagnetism of spin-12 ions

The spin is either up or down relative to the field, and so the magneticmoment is either+µB or −µB, where

µB =e~

2me= 9.274× 10−24 Am2.

The corresponding energies in a flux densityB are −µBB and µBB,so the average magnetic moment per atom is

〈µ〉 =µBeµBB/kBT − µBe−µBB/kBT

eµBB/kBT + e−µBB/kBT

= µB tanh

(µBBkBT

).

For small z, tanh z ≈ z, so for small fields or high temperature

〈µ〉 ≈µ2

BBkBT

.

If there are n atoms per volume, then,

χ =nµ0µ

2B

kBT.

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Clearly, though, for low T or large B the magnetic moment per atomsaturates – as it must, as the largest magnetisation possiblesaturationmagnetisationhas all the spins aligned fully,

Ms = nµB.

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